The answer is negative. To construct a counterexample I used Jan Maly's paper THE DARBOUX PROPERTY FOR GRADIENTS, 1996.
First let's consider
$$
\varphi(x,y):= \begin{cases} \displaystyle \frac {2xy^4}{x^2+y^4} -x, (x,y) \ne(0,0); \\0,(x,y)=(0,0). \end{cases}
$$
We observe that $\varphi(x,y)$ is everywhere differentiable, $\varphi^\prime_x(0,0)=-1$, $\varphi^\prime_x(0,y)=1 $ $\forall y\ne0$ and $\varphi^\prime_y(0,y)=0$. Now let's consider $f(x,y):=(\varphi(x,y),x+y)$. It's clear that $f:R^2\to R^2$ is everywhere differentiable and $$
J_f(0,y) = \begin{cases} 1, y\ne0; \\-1, y=0. \end{cases}
$$
Thus intermediate value (Darboux) property for Jacobian determinant doesn't hold even on a straight line $x=0$.

isan intermediate value theorem, even without the assumption that it's continuous. (Don't remember the exact name.) So, the question about the Jacobian may be not as simple as it seems. $\endgroup$ – Alex Degtyarev Mar 28 '14 at 8:52